Time dilation is the effect that for some objects, less time as passed than for others, depending their position and velocity. It is a consequence of relativity. Special relativity implies time dilation based upon the relative motion of two objects. General relativity also produces time dilation (gravitational time dilation) for objects deeper within a gravitational well compared to those not so deep within it. Time dilation challenges intuition based upon daily experience, but is not only proven, but some technology that depends upon high-precision time measurement, such as GPS, requires time dilation to be taken into account in order to function. It is not illusion: two accurate clocks go out of sync if one of them is moved away, then back to the other.
Cosmological time dilation is the apparent time dilation of events viewed from large (high redshift) distances.